Computing interval parameter bounds from fallible measurements using systems of nonlinear equations

ABSTRACT

One embodiment of the present invention provides a system that computes interval parameter bounds from fallible measurements. During operation, the system receives a set of measurements z 1   , . . . , z   n , wherein an observation model describes each z 1  as a function of a p-element vector parameter x=(x i , . . . , x p ). Next, the system forms a system of nonlinear equations z i −h(x)=0 (i=1, . . . , n) based on the observation model. Finally, the system solves the system of nonlinear equations to determine interval parameter bounds on x.

RELATED APPLICATIONS

[0001] This application hereby claims priority under 35 U.S.C. §119 toU.S. Provisional Patent Application No. 60/396,246, filed on Jul. 16,2002, entitled, “Overdetermined (Tall) Systems of Nonlinear Equations,”by inventors G. William Walster and Eldon R. Hansen (Attorney Docket No.SUN-8507PSP).

BACKGROUND

[0002] 1. Field of the Invention

[0003] The present invention relates to techniques for performingarithmetic operations involving interval operands within a computersystem. More specifically, the present invention relates to a method andan apparatus for computing interval parameter bounds from falliblemeasurements using systems of nonlinear equations.

[0004] 2. Related Art

[0005] Rapid advances in computing technology make it possible toperform trillions of computational operations each second. Thistremendous computational speed makes it practical to performcomputationally intensive tasks as diverse as predicting the weather andoptimizing the design of an aircraft engine. Such computational tasksare typically performed using machine-representable floating-pointnumbers to approximate values of real numbers. (For example, see theInstitute of Electrical and Electronics Engineers (IEEE) standard 754for binary floating-point numbers.)

[0006] In spite of their limitations, floating-point numbers aregenerally used to perform most computational tasks.

[0007] One limitation is that machine-representable floating-pointnumbers have a fixed-size word length, which limits their accuracy. Notethat a floating-point number is typically encoded using a 32, 64 or128-bit binary number, which means that there are only 2³², 2⁶⁴ or 2¹²⁸possible symbols that can be used to specify a floating-point number.Hence, most real number values can only be approximated with acorresponding floating-point number. This creates estimation errors thatcan be magnified through even a few computations, thereby adverselyaffecting the accuracy of a computation.

[0008] A related limitation is that floating-point numbers contain noinformation about their accuracy. Most measured data values include someamount of error that arises from the measurement process itself. Thiserror can often be quantified as an accuracy parameter, which cansubsequently be used to determine the accuracy of a computation.However, floating-point numbers are not designed to keep track ofaccuracy information, whether from input data measurement errors ormachine rounding errors. Hence, it is not possible to determine theaccuracy of a computation by merely examining the floating-point numberthat results from the computation.

[0009] Interval arithmetic has been developed to solve theabove-described problems. Interval arithmetic represents numbers asintervals specified by a first (left) endpoint and a second (right)endpoint. For example, the interval [a, b], where a<b, is a closed,bounded subset of the real numbers, R, which includes a and b as well asall real numbers between a and b. Arithmetic operations on intervaloperands (interval arithmetic) are defined so that interval resultsalways contain the entire set of possible values. The result is amathematical system for rigorously bounding numerical errors from allsources, including measurement data errors, machine rounding errors andtheir interactions. (Note that the first endpoint normally contains the“infimum”, which is the largest number that is less than or equal toeach of a given set of real numbers. Similarly, the second endpointnormally contains the “supremum”, which is the smallest number that isgreater than or equal to each of the given set of real numbers. Alsonote that the infimum and the supremum can be represented by floatingpoint numbers.)

[0010] One commonly performed operation is to compute bounds onnonlinear parameters from a set of fallible measurements. Using thetraditionally accepted methodology to compute approximate parametervalues from nonlinear models of observable data requires a number ofquestionable assumptions. In the best case, if all assumptions aresatisfied, the final result is a less than 100% statistical confidenceinterval rather than a containing interval bound. For example, themethod of least squares produces a solution approximation even when thedata on which it is based are inconsistent.

[0011] Hence, what is needed is a method and an apparatus that usesinterval techniques to compute bounds on nonlinear parameters fromfallible measurements.

SUMMARY

[0012] One embodiment of the present invention provides a system thatcomputes interval parameter bounds from fallible measurements. Duringoperation, the system receives a set of measurements z₁, . . . , z_(n)wherein an observation model describes each z_(i) as a function of ap-element vector parameter x=(x₁, . . . , x_(p)). Next, the system formsa system of nonlinear equations z_(i−h(x)=)0 (i=1, . . . , n) based onthe observation model. Finally, the system solves the system ofnonlinear equations to determine interval parameter bounds on x.

[0013] In a variation on this embodiment, the system of nonlinearequations is an “overdetermined system” in which there are moreequations than unknowns.

[0014] In a variation on this embodiment, each measurement z_(i) isactually a q-element vector of measurements z_(i)=(z_(il), . . . ,z_(iq))^(T), and h is actually a q-element vector of functions h=(h₁, .. . , h_(q))^(T).

[0015] In a variation on this embodiment, receiving the set ofmeasurements involves receiving values for a set of conditions c₁, . . ., c_(n) under which the corresponding observations z_(i) were made. Inthis variation, the system of nonlinear equations is of the formz_(i−h(x|c) _(i))=0 (i=1, . . . , n).

[0016] In a further variation, each condition c_(i) is actually anr-element vector of conditions c_(i)=(c_(il), . . . , c_(ir))^(T).

[0017] In a further variation, each condition c_(i) is not knownprecisely but is contained within an interval c^(I) _(i).

[0018] In a variation on this embodiment, equations in the system ofnonlinear equations are of the form z_(i)−h(x|c_(i))+ε^(I)(x, c_(i))=0(i=1, . . . , n), which includes an error model ε^(I)(x, c_(i)) thatprovides interval bounds on measurement errors for z_(i).

[0019] In a further variation, if z_(i) is actually a q-element vectorof measurements z_(i)=(z_(il), . . . , z_(iq))^(T), then ε^(I) isactually a q-element vector ε^(I)=(ε₁, . . . , ε_(q))^(T).

[0020] In a further variation, if there exists no solution to the systemof nonlinear equations, the system determines that at least one of thefollowing is true: (1) at least one of the set of measurements z_(i), .. . , z_(n) is faulty; (2) the observation model h(x|c_(i)) is false;(3) the error model ε^(I)(x, c_(i)) is false; and (4) the computationalsystem used to compute interval bounds on elements of x is flawed.

[0021] In a variation on this embodiment, solving the system ofnonlinear equations involves first linearizing the system of nonlinearequations to form a corresponding system of linear equations, and thensolving the system of linear equations through Gaussian elimination.

BRIEF DESCRIPTION OF THE FIGURES

[0022]FIG. 1 illustrates a computer system in accordance with anembodiment of the present invention.

[0023]FIG. 2 illustrates the process of compiling and using code forinterval computations in accordance with an embodiment of the presentinvention.

[0024]FIG. 3 illustrates an arithmetic unit for interval computations inaccordance with an embodiment of the present invention.

[0025]FIG. 4 is a flow chart illustrating the process of performing aninterval computation in accordance with an embodiment of the presentinvention.

[0026]FIG. 5 illustrates four different interval operations inaccordance with an embodiment of the present invention.

[0027]FIG. 6 illustrates the process of performing a GaussianElimination operation on an overdetermined interval system of linearequations in accordance with an embodiment of the present invention.

[0028]FIG. 7 illustrates the process of generating a preconditioningmatrix in accordance with an embodiment of the present invention.

[0029]FIG. 8 presents a flow chart illustrating the process of computinginterval parameter bounds from fallible measurements in accordance withan embodiment of the present invention.

[0030] Table 1 (located near the near the end of the specification—notwith the figures) illustrates a correspondence between parameterestimation and nonlinear equations in accordance with an embodiment ofthe present invention.

DETAILED DESCRIPTION

[0031] The following description is presented to enable any personskilled in the art to make and use the invention, and is provided in thecontext of a particular application and its requirements. Variousmodifications to the disclosed embodiments will be readily apparent tothose skilled in the art, and the general principles defined herein maybe applied to other embodiments and applications without departing fromthe spirit and scope of the present invention. Thus, the presentinvention is not intended to be limited to the embodiments shown, but isto be accorded the widest scope consistent with the principles andfeatures disclosed herein.

[0032] The data structures and code described in this detaileddescription are typically stored on a computer readable storage medium,which may be any device or medium that can store code and/or data foruse by a computer system. This includes, but is not limited to, magneticand optical storage devices such as disk drives, magnetic tape, CDs(compact discs) and DVDs (digital versatile discs or digital videodiscs), and computer instruction signals embodied in a transmissionmedium (with or without a carrier wave upon which the signals aremodulated). For example, the transmission medium may include acommunications network, such as the Internet.

[0033] Computer System

[0034]FIG. 1 illustrates a computer system 100 in accordance with anembodiment of the present invention. As illustrated in FIG. 1, computersystem 100 includes processor 102, which is coupled to a memory 112 anda to peripheral bus 110 through bridge 106. Bridge 106 can generallyinclude any type of circuitry for coupling components of computer system100 together.

[0035] Processor 102 can include any type of processor, including, butnot limited to, a microprocessor, a mainframe computer, a digital signalprocessor, a personal organizer, a device controller and a computationalengine within an appliance. Processor 102 includes an arithmetic unit104, which is capable of performing computational operations usingfloating-point numbers. Processor 102 communicates with storage device108 through bridge 106. and peripheral bus 110. Storage device 108 caninclude any type of non-volatile storage device that can be coupled to acomputer system. This includes, but is not limited to, magnetic,optical, and magneto-optical storage devices, as well as storage devicesbased on flash memory and/or battery-backed up memory.

[0036] Processor 102 communicates with memory 112 through bridge 106.Memory 112 can include any type of memory that can store code and datafor execution by processor 102. As illustrated in FIG. 1, memory 112contains computational code for intervals 114. Computational code 114contains instructions for the interval operations to be performed onindividual operands, or interval values 115, which are also storedwithin memory 112. This computational code 114 and these interval values115 are described in more detail below with reference to FIGS. 2-5.

[0037] Note that although the present invention is described in thecontext of computer system 100 illustrated in FIG. 1, the presentinvention can generally operate on any type of computing device that canperform computations involving floating-point numbers. Hence, thepresent invention is not limited to the computer system 100 illustratesin FIG. 1.

[0038] Compiling and Using Interval Code

[0039]FIG. 2 illustrates the process of compiling and using code forinterval computations in accordance with an embodiment of the presentinvention. The system starts with source code 202, which specifies anumber of computational operations involving intervals. Source code 202passes through compiler 204, which converts source code 202 intoexecutable code form 206 for interval computations. Processor 102retrieves executable code 206 and uses it to control the operation ofarithmetic unit 104.

[0040] Processor 102 also retrieves interval values 115 from memory 112and passes these interval values 115 through arithmetic unit 104 toproduce results 212. Results 212 can also include interval values.

[0041] Note that the term “compilation” as used in this specification isto be construed broadly to include pre-compilation and just-in-timecompilation, as well as use of an interpreter that interpretsinstructions at run-time. Hence, the term “compiler” as used in thespecification and the claims refers to pre-compilers, just-in-timecompilers and interpreters.

[0042] Arithmetic Unit for Intervals

[0043]FIG. 3 illustrates arithmetic unit 104 for interval computationsin more detail accordance with an embodiment of the present invention.Details regarding the construction of such an arithmetic unit are wellknown in the art. For example, see U.S. Pat. Nos. 5,687,106 and6,044,454. Arithmetic unit 104 receives intervals 302 and 312 as inputsand produces interval 322 as an output.

[0044] In the embodiment illustrated in FIG. 3, interval 302 includes afirst floating-point number 304 representing a first endpoint ofinterval 302, and a second floating-point number 306 representing asecond endpoint of interval 302. Similarly, interval 312 includes afirst floating-point number 314 representing a first endpoint ofinterval 312, and a second floating-point number 316 representing asecond endpoint of interval 312. Also, the resulting interval 322includes a first floating-point number 324 representing a first endpointof interval 322, and a second floating-point number 326 representing asecond endpoint of interval 322.

[0045] Note that arithmetic unit 104 includes circuitry for performingthe interval operations that are outlined in FIG. 5. This circuitryenables the interval operations to be performed efficiently.

[0046] However, note that the present invention can also be applied tocomputing devices that do not include special-purpose hardware forperforming interval operations. In such computing devices, compiler 204converts interval operations into a executable code that can be executedusing standard computational hardware that is not specially designed forinterval operations.

[0047]FIG. 4 is a flow chart illustrating the process of performing aninterval computation in accordance with an embodiment of the presentinvention. The system starts by receiving a representation of aninterval, such as first floating-point number 304 and secondfloating-point number 306 (step 402). Next, the system performs anarithmetic operation using the representation of the interval to producea result (step 404). The possibilities for this arithmetic operation aredescribed in more detail below with reference to FIG. 5.

[0048] Interval Operations

[0049]FIG. 5 illustrates four different interval operations inaccordance with an embodiment of the present invention. These intervaloperations operate on the intervals X and Y. The interval X includes twoendpoints,

[0050]x denotes the lower bound of X, and

[0051] {overscore (x)} denotes the upper bound of X.

[0052] The interval X is a closed subset of the extended (including −∞and +∞) system of real numbers R* (see line 1 of FIG. 5). Similarly theinterval Y also has two endpoints and is a closed subset of the extendedreal numbers R* (see line 2 of FIG. 5).

[0053] Note that an interval is a point or degenerate interval if X=[x,x]. Also note that the left endpoint of an interior interval is alwaysless than or equal to the right endpoint. The set of extended realnumbers, R* is the set of real numbers, R, extended with the two idealpoints negative infinity and positive infinity:

R*=(R∪{−∞})∪{+∞}=[−∞,+∞].

[0054] We also define R** by replacing the unsigned zero, {0}, from R*with the interval [−0,+0].

R**=R*−{0}∪[−0,+0]=[−∞,+∞], because 0=[−0,+0].

[0055] In the equations that appear in FIG. 5, the up arrows and downarrows indicate the direction of rounding in the next and subsequentoperations. Directed rounding (up or down) is applied if the result of afloating-point operation is not machine-representable.

[0056] The addition operation X+Y adds the left endpoint of X to theleft endpoint of Y and rounds down to the nearest floating-point numberto produce a resulting left endpoint, and adds the right endpoint of Xto the right endpoint of Y and rounds up to the nearest floating-pointnumber to produce a resulting right endpoint.

[0057] Similarly, the subtraction operation X−Y subtracts the rightendpoint of Y from the left endpoint of X and rounds down to produce aresulting left endpoint, and subtracts the left endpoint of Y from theright endpoint of X and rounds up to produce a resulting right endpoint.

[0058] The multiplication operation selects the minimum value of fourdifferent terms (rounded down) to produce the resulting left endpoint.These terms are: the left endpoint of X multiplied by the left endpointof Y; the left endpoint of X multiplied by the right endpoint of Y; theright endpoint of X multiplied by the left endpoint of Y; and the rightendpoint of X multiplied by the right endpoint of Y. This multiplicationoperation additionally selects the maximum of the same four terms(rounded up) to produce the resulting right endpoint.

[0059] Similarly, the division operation selects the minimum of fourdifferent terms (rounded down) to produce the resulting left endpoint.These terms are: the left endpoint of X divided by the left endpoint ofY; the left endpoint of X divided by the right endpoint of Y; the rightendpoint of X divided by the left endpoint of Y; and the right endpointof X divided by the right endpoint of Y. This division operationadditionally selects the maximum of the same four terms (rounded up) toproduce the resulting right endpoint. For the special case where theinterval Y includes zero, X/Y is an exterior interval that isnevertheless contained in the interval R*.

[0060] Note that the result of any of these interval operations is theempty interval if either of the intervals, X or Y, are the emptyinterval. Also note, that in one embodiment of the present invention,extended interval operations never cause undefined outcomes, which arereferred to as “exceptions” in the IEEE 754 standard.

[0061] Solving an Overdetermined System of Interval Linear Equations

[0062] In order to solve a system of interval nonlinear equations, wefirst describe a technique for solving a system of interval linearequations. We can subsequently use this technique in solving acorresponding system of interval nonlinear equations.

[0063] Given the real (n×n) matrix A and the (n×l) column vector b, thelinear system of equations

Ax=b  (1)

[0064] is consistent if there is a unique (n×l) vector x for which thesystem in (1) is satisfied. If the number of rows in A and elements in bis m≠n, then the system is said to be either under- or overdetermineddepending on whether m<n or n<m. In the overdetermined case, if m−nequations are not linearly dependent on the remaining equations, thereis no solution vector x that satisfies the system. In theunderdetermined case there is no unique solution.

[0065] In the point (non-interval) case, there is no generally reliableway to decide if an overdetermined system based on fallible observationsis consistent or not. Instead a least squares solution is generallysought. In the interval case, if the system of equations is sufficientlyinconsistent, the computed interval solution set will be empty. If thereare at least some parameter values that are consistent with all theobservations, it is possible to delete inconsistent parameter values andbound the consistent ones.

[0066] We now consider the problem of solving overdetermined systems ofequations in which the coefficients are intervals. That is, we considera system of the form

A^(I)x=b^(I)  (2)

[0067] where A^(I) is an interval matrix of m rows and n columns withm>n. The interval vector b^(I) has m components. Such a system mightarise directly or by linearizing an overdetermined system of nonlinearequations. (Note that within this specification and in the followingclaims, we sometimes drop the superscript “I” when referring theinterval matrices or vectors.)

[0068] The solution set of (2) is the set of vectors x for which thereexists a real matrix A ε A^(I) and a real vector b ε b^(I) such that (1)is satisfied. In general, the system in (2) is inconsistent if itssolution set is empty. First, we assume that there exists at least one Aε A^(I) and b ε b^(I) such that (1) is consistent. Later, we considerthe inconsistent case. Moreover, we also assume that the data in A^(I)and b^(I) are fallible. That is, there exists at least one A ε A^(I) andb ε b^(I) such that (1) is inconsistent. Our goal is to implicitlyexclude values of x that are inconsistent with all A ε A^(I) and b εb^(I). For example, the redundancy resulting from the fact that thereare more equations than variables might be deliberately introduced tosharpen the interval bound on the set of solutions to (2). In afollowing section, we show how this sharpening is accomplished.

[0069] We shall simplify the system using Gaussian elimination. In thepoint case, it is good practice to avoid forming normal equations fromthe original system. Instead, one performs elimination using normaloperation matrices to zero all elements of the coefficient matrix exceptfor an upper triangle. After this first phase, the normal equations ofthis simpler system can be formed and solved. Our procedure begins witha phase similar to the first phase just described. However, we do notquite complete the usual elimination procedure. We have no motivation touse normal operations because we do not form the normal equations. Thisis just as well because interval normal matrices do not exist.

[0070] When using interval Gaussian elimination, it is generallynecessary to precondition the system to avoid excessive widening ofintervals due to dependence. In the following section, we show howpreconditioning can be done in the present case where A^(I) is notsquare.

[0071] Preconditioning

[0072] Preconditioning can be done in the same way it is done when A^(I)is square. Let A_(c) denote the center of the interval matrix A^(I).Partition A_(c) as $\begin{matrix}{A_{c} = \begin{bmatrix}A_{c}^{\prime} \\A_{c}^{''}\end{bmatrix}} & (3)\end{matrix}$

[0073] where A′_(c) is an n by n matrix and A″_(c) ″is an m−n by nmatrix. Note that A_(c) need only be an approximation for the center ofA^(I). Define the partitioned square matrix $\begin{matrix}{C = \begin{bmatrix}A_{c}^{\prime} & 0 \\A_{c}^{''} & I\end{bmatrix}} & (4)\end{matrix}$

[0074] where I denotes the identity matrix of order m−n, and the blockdenoted by 0 is an n×m−n matrix of zeros.

[0075] Define the preconditioning matrix B to be an approximation forthe inverse $\begin{bmatrix}\left( A_{c}^{\prime} \right)^{- 1} & 0 \\{A_{c}^{''}\left( A_{c}^{\prime} \right)}^{- 1} & I\end{bmatrix}\quad$

[0076] of C.

[0077] To precondition (2) we multiply by B. We obtain

M^(I)x=r^(I)  (5)

[0078] where M^(I)=BA^(I) is an m by n interval matrix and r^(I)=Bb^(I)is an interval vector of m components. When computing M^(I) and r^(I),we use interval arithmetic to bound rounding errors.

[0079] Elimination

[0080] We now perform elimination. We apply an interval version ofGaussian elimination to the system M^(I)x=r^(I) thereby transformingM^(I) into almost (see below) upper trapezoidal form. We assume thatthis procedure only fails when all possible pivot elements contain zero.Note that after preconditioning, no pivot selection is performed duringthe elimination to obtain a result with the form $\begin{matrix}{{\begin{bmatrix}T^{I} \\W^{I}\end{bmatrix}x} = \begin{bmatrix}u^{I} \\v^{I}\end{bmatrix}} & (6)\end{matrix}$

[0081] where T^(I) is a square upper triangular interval matrix of ordern, and both u^(I) and v^(I) are interval vectors of n and m−ncomponents, respectively. The submatrix W^(I) is a matrix of m−n rowsand n columns. It is zero except in the last column. Therefore, we canrepresent it in the form

W ^(I)=[0z ^(I])

[0082] where 0 denotes an m−n by n−1 block of zeros, and z^(I) is avector of m−n intervals. From (6), we now have a set of equations

z _(i) x _(n) =v _(i)(i=1, . . . , m−n).  (7)

[0083] Also,

T_(nn)x_(n)=u_(n).  (8)

[0084] Therefore, the unknown value x_(n) is contained in the interval$\begin{matrix}{x_{n} = {\frac{u_{n}}{T_{nn}}\bigcap\limits_{i = 1}^{m - n}{\frac{v_{i}}{z_{i}}.}}} & (9)\end{matrix}$

[0085] Taking this intersection is what implicitly eliminates fallibledata from A^(I) and b^(I). It is this operation that allows us to get asharper bound on the set of solutions to the original system (2) thanmight otherwise be obtained.

[0086] If the original system contains at least one consistent set ofequations, the intersection in (9) will not be empty. Knowing x_(n) wecan backsolve (6) for x_(n−1), . . . , x_(i). From (6), this takes thestandard form of backsolving a triangular system T^(I)x=u^(I).Sharpening x_(n) using (9) also produces sharper bounds x^(I) on theother components of x when we backsolve.

[0087] Inconsistency

[0088] Now suppose the initial equations (2) are not consistent. Thenthe preconditions of equations (7) might or might not be consistent.Widening of intervals due to dependence and roundoff can cause theintersection in (9) to be non-empty.

[0089] Nevertheless, suppose we find that the intersection in (9) isempty. This event proves that the original equations (2) areinconsistent. Proving inconsistency might be the signal that a theory ismeasurably false, which might be an extremely enlightening event. On theother hand, inconsistency might only mean that invalid measurements havebeen made.

[0090] If invalid measurements are suspected, it might be important todiscover which equation(s) in (2) are inconsistent. We might know whichequation(s) in the transformed system (6) must be eliminated to obtainconsistency. However, an equation in (6) is generally a linearcombination of all the original equations in (2). Therefore, toestablish consistency in the original system, we generally cannotdetermine which of its equation(s) to remove.

[0091] We might be able to determine a likely removal candidate by usingthe following steps:

[0092] 1. Remove enough equations from (6) that the intersection in (9)is not empty.

[0093] 2. Solve (6) for x_(n−1), . . . , x₁. This process cannot failbecause we assume the elimination process to obtain (6) does not fail.

[0094] 3. Substitute the solution into the original system (2). Anyequation(s) in (2) whose left and right members do not intersect can bediscarded.

[0095] Summary of the Gaussian Elimination Operation

[0096]FIG. 6 illustrates the process of performing a GaussianElimination operation on an overdetermined interval system of linearequations in accordance with an embodiment of the present invention. Thesystem starts by receiving a representation of the overdetermined systemof linear equations Ax=b (step 602). In this representation, A is amatrix with m rows corresponding to m equations and n columnscorresponding to n variables, x includes n variable components, bincludes m scalar components, and m>n. The system then stores thisrepresentation in memory (step 604).

[0097] Next, the system preconditions Ax=b to generate a modified systemBAx=Bb that can be solved with reduced growth of interval widths (step606). This preconditioning process is described in more detail belowwith reference to FIG. 7.

[0098] The system then performs an interval Gaussian eliminationoperation on BAx=Bb to form ${{\begin{bmatrix}T \\W\end{bmatrix}x} = \begin{bmatrix}u \\v\end{bmatrix}},$

[0099] wherein T is a square upper triangular matrix of order n, u is aninterval vector with n components, v is an interval vector with m−ncomponents, and W is a matrix with m−n rows and n columns, and wherein Wis zero except in the last column, which is represented as a columnvector z with m−n components (step 608).

[0100] Note that interval Gaussian elimination can fail. If so, thesystem simply terminates (step 609).

[0101] If Gaussian elimination does not fail, the system performs aninterval intersection operation based on the equationsz_(i)x_(n)=v_(i)(i=1, . . . ,m−n) and T_(nn)x_(n)=u_(n) to solve for$x_{n} = {\frac{u_{n}}{T_{nn}}\bigcap\limits_{i = 1}^{m - n}\frac{v_{i}}{z_{i}}}$

[0102] (step 610).

[0103] Finally, if x_(n) is not the empty interval, the system performsa back substitution operation using x_(n) and Tx=u to solve for theremaining components (x_(n−1), . . . , x₁) of x (step 612).

[0104]FIG. 7 illustrates the process of generating a preconditioningmatrix in accordance with an embodiment of the present invention. Thesystem starts by determining a non-interval matrix A_(c), which is theapproximate center of the interval matrix A (step 702). Next, the systemaugments the m×n matrix A_(c) to produce an n×n partitioned matrix${C = \begin{bmatrix}A_{c}^{\prime} & 0 \\A_{c}^{''} & I\end{bmatrix}},$

[0105] wherein A′_(c) is an n×n matrix, A″_(c) is an m−n×n matrix, I isthe identity matrix of order m−n, and 0 is an n×m−n matrix of zeros(step 704). Finally, the system calculates the approximate inverse ofthe partitioned matrix C to produce the preconditioning matrix B (step706). If C happens to be singular, its elements can be perturbed untilit is no longer so. This causes no difficulty because C is just used tocompute the approximate inverse B.

[0106] Parameter Estimation in Nonlinear Models

[0107] Overdetermined (tall) systems of nonlinear equations naturallyarise in the context of computing interval parameter bounds fromfallible data. In tall systems, there are more interval equations thanunknowns. As a result, these systems can appear to be inconsistent whenthey are not. A technique is described to compute interval nonlinearparameter bounds from fallible data and to possibly prove that no boundsexist because the tall system is inconsistent.

[0108] Interval arithmetic has been used to perform the analysis offallible observations from an experiment to compute bounds on Newton'sconstant of gravitation G. (see B. Lang, Verified Quadrature inDetermining Newton's Constant of Gravitation, Journal of UniversalComputer Science, 4(1):16-24, 1998.) Because the computed bounds weresufficiently different from the then accepted approximate value,subsequent experiments were conducted to refine the accepted approximatevalue and the interval bound on G. (see “The Controversy of Newton'sGravitational Constant,” The Eöt-Wash Group: Laboratory of GravitationalPhysics, www.npl.washington.edu/eotwash/gconst.html)

[0109] Using the traditionally accepted methodology to computeapproximate parameter values from nonlinear models of observable datarequires a number of questionable assumptions. In the best case, if allassumptions are satisfied, the final result is a less than 100%statistical confidence interval rather than a containing interval bound.For example, the method of least squares produces a solutionapproximation even when the data on which it is based are inconsistent.

[0110] A better procedure is to solve a system of interval nonlinearequations using the interval version of Newton's method. If assumptionsfor this procedure are satisfied, the result is a guaranteed bound onthe parameter(s) in question. If assumptions are sufficiently violatedand enough observations are available, the procedure can prove thesystem of equations and interval data are inconsistent. This betterprocedure is now described.

[0111] Nonlinear Parameter Estimation

[0112] Let n q-element vector measurements, z₁, . . . , z_(n) withz_(i)=(z_(il), . . . , z_(iq))^(T) be given. Assume these measurementsdepend on the value of a p-element vector parameter, x_(i)=(x₁, . . . ,x_(p))^(T). Moreover, assume an analytic model exists for theobservation vectors, z_(i), as a function of x and the true valuec_(i)=(C_(il), . . . , c_(ir))^(T) of conditions under which the z_(i)are measured. Thus:

z _(i) =h(x|c _(i))  (11)

[0113] The problem is to construct interval bounds x^(I) on the elementsof x from interval bounds z^(I) _(i) on the fallible measurements, z_(i)and interval bounds c^(I) _(i) on the conditions of measurement.

[0114] Interval Observation Bounds

[0115] The development of interval measurement bounds begins byrecognizing that a measurement z can be modeled (or thought of) as anunknown value t to which an error is added from the interval

ε×[−1,1]=ε^(I)  (12)

[0116] where 0≦ε. No assumption is made about the distribution ofindividual measurement errors from the interval ε^(I) that are added tot in the process of measuring z.

[0117] At once it follows that

zεt+ε^(I).  (13)

[0118] More importantly, if the interval observation Z is defined to bez+ε^(I), then

tεZ.  (14)

[0119] Enclosure (14) is an immediate consequence of the fact that zerois the midpoint of ε^(I). This simple idea has a number of implications.They are:

[0120] Given multiple interval observations Z_(i), all of which areenclosures of t, so must their intersection. Therefore$t \in {\bigcap\limits_{i = 1}^{n}{Z_{i}.}}$

[0121] Given random finite intervals Z_(i), all of which contain thevalue t, the expected width of their intersection decreases as nincreases.

[0122] An empty intersection is proof that t ∉ Z_(i) for some value ofi. This can be true either because

[0123] the width of the interval observations Z_(i) is too narrow,

[0124] there is no single value t that is contained in all the intervalmeasurements Z_(i), or

[0125] both of the above.

[0126] The first alternative means that the assumption regarding theaccuracy of the measurement process is false. The second alternativemeans that the model for the single common value t is false.

[0127] Walster explored how this simple idea works in practice tocompute an interval bound on a common value under various probabilitydistributions for values of the random variable ε ε ε^(I). (see G. W.Walster, “Philosophy and Practicalities of Interval Arithmetic,” R. E.Moore Editor, Reliability on Computing, pages 309-323, Academic Press,Inc., San Diego, Calif. 1988.) He also discussed how this estimationprinciple can be generalized and used to bound parameters of nonlinearmodels given bounded interval observations, or observation vectors. Thefollowing is a more complete elaboration of the nonlineargeneralization.

[0128] General Development

[0129] Given exact values of a set of conditions c_(i)=(c_(il), . . . ,c_(ir))^(T) under which observations z_(i)=(z_(il), . . . , z_(iq))^(T)are made, assume the observation vectors, z_(i) (i=1, . . . , n),satisfy the following model:

z_(i)εh(x|c_(i))+ε(x, c_(i))×[−1,1];  (15)

[0130] where the vectors 0≦ε(x, c_(i)) bound unknown modeling and directmeasurement errors. Specifically, if the p elements of x and the rnelements of all the c_(i) were known (in practice they are not), assumeit would be possible to compute intervals

ε^(I)(x, c _(i))=ε(x, c _(i))×[−1,1];  (16)

[0131] from which it follows immediately that

0εz_(i)−h(x|c_(i))+ε^(I)(x, c_(i)).  (17)

[0132] Note that (16) is a generalization of (12), and (17) is ageneralization of (14) if written in the form 0 ε Z−t.

[0133] If (as is normally the case) the conditions c_(i) under whichmeasurements z_(i) are made are not known, but are contained inintervals c^(I) _(i), then taking all bounded modeling and observationerrors into account:

0εz_(i)−h(x|c^(I) _(i))+ε^(I)(x, c^(I) _(i)).  (18)

[0134] In this general form, the widths of interval measurements z_(I)_(i) are themselves functions of both the unknown parameters x andfallibly measured conditions under which the measurements are made. Thatis:

z ^(I) _(i) =z _(i)+ε^(I)(x, c ^(I) _(i)).  (19)

[0135] This interval observation model is the generalization ofZ=z+ε^(I) which is consistent with (13). The interval observation model(19) is needed to solve for interval bounds on the parameter vector x.If there is no solution for a given set of interval observation vectorsz^(I) _(i) and interval bounds on measurement conditions c^(I) _(i),then either:

[0136] the observation model h(x|c^(I) _(i)) is false;

[0137] the measurement error model ε^(I)(x, c^(I) _(i)) is false;

[0138] the computational system used to compute interval bounds on theelements of x is flawed; or

[0139] some combination of the above.

[0140] In this way, and by eliminating alternative explanations, thetheory represented in h(x|c^(I) _(i)) or the observation error modelrepresented in ε^(I)(x, c^(I) _(i)) can be proved to be false.

[0141] The System of Nonlinear Equations

[0142] To guarantee any computed interval x^(I) is indeed a valid boundon the true value of x, the following must be true:

[0143] the given model h;

[0144] the interval bounds c^(I) _(i) on the conditions under whichfallible measurements z_(i) are made; and,

[0145] the model for interval bounds ε^(I)(x, c^(I) _(i)) on observationerrors.

[0146] To be consistent with the given models, all the actualmeasurement vectors z_(i) must satisfy relation (18). A logicallyequivalent, but more suggestive way to write this system of constraintsis:

z _(i) −h(x|c ^(I) _(i))+ε^(I)(x, c ^(I) _(i))=0(i=1, . . . , n)  (20)

[0147] When used in (20), a possible value of x produces intervals thatcontain zero for all i. Any value of x that fails to do this cannot bein the solution set of (20). Thus, (20) is just an interval system ofnonlinear equations in the unknown parameter vector x. The problem isthat the total number of scalar equations nq might be much larger thanthe number p of scalar unknowns in the parameter vector x. Point (ratherthan interval) systems of equations where p<nq are called“overdetermined”. For interval nonlinear equations, this is a misnomerbecause the interval equations might or might not be consistent. Asmentioned above, inconsistency (an empty solution set) is an informativeevent.

[0148] Solving Nonlinear Equations

[0149] Let f:

^(n)→

^(m) (n≦m) be a continuously differentiable function. The parameterestimation problem described above is just a special case of the moregeneral problem now considered. Table 1 below shows the correspondencebetween the parameter estimation problem and equivalent nonlinearequations to be solved. Both unknowns and equations are shown. TABLE 1Correspondence between Parameter Estimation and Nonlinear EquationsParameter Estimation Nonlinear Equations Unknownsx^(I) = (x₁^(I), …  , x_(p)^(I))^(T)

x^(I) = (X₁, …  , X_(p))^(T)

Equationsz_(i) − h(x|c_(i)^(I)) + ɛ^(I)(x, c_(i)^(I)) = 0  (i = 1, …  , n)

f = (f₁, …  , f_(m))^(T) = 0  where  m = nq

[0150] Having established this correspondence, the problem becomes tofind and bound all the solution vectors of f(x)=0 in a given initial boxx^(I(0)). For non-interval methods, it can sometimes be difficult tofind reasonable bounds on a single solution, quite difficult to findreasonable bounds on all solutions, and generally impossible to knowwhether reasonable bounds on all solutions have been found. In contrast,it is a straightforward problem to find reasonable bounds on solutionsin x^(I(0)) using interval methods; and it is trivially easy tocomputationally determine that all solutions in x^(I(0)) have beenbounded. What is unusual in this problem is that the order m of f can begreater than the order p of x^(I). A factor that simplifies obtainingsolution(s) is the assumption that the equations are consistent. Thishas the effect of reducing the number of equations at a solution to thenumber of variables.

[0151] Linearization and Gaussian Elimination

[0152] Let x and y be points in a box x^(I). Suppose we expand eachcomponent ƒ_(i) (i=1, . . . , m) of f by one of the procedures commonlyused to linearize nonlinear equations to be solved using the intervalNewton method. Define the matrix of partial derivatives of the elementsƒ_(i) of f with respect to the elements x_(j) (j=1, . . . , p) of x:$J_{ij} = {\left( \frac{\partial{f_{i}(x)}}{\partial x_{j}} \right).}$

[0153] If n=m, the system is square, and J is the Jacobian of f. This isthe usual situation in which the interval Newton method is applied. (see[Hansen] E. R. Hansen, “Global Optimization Using Interval Analysis,”Marcel Dekker, Inc., New York, 1992).

[0154] In passing it is worth noting that in place of partialderivatives, slopes can be used to good advantage. Slopes have narrowerwidth than interval bounds on derivatives and might exist whenderivatives are undefined. Nevertheless, the remaining development usesderivatives as they are more familiar than slopes.

[0155] Combining the results in vector form:

f(y)εf(x)+J(x, x^(I))(y−x)  (21)

[0156] Even in the non-square situation, J is still referred to hereinas the Jacobian of f. The notation J(x, x^(I)) is used to emphasize thefact that a tighter expansion of ƒ can be obtained if both point andinterval values of x elements are used to compute Jacobian matrixelements (see [Hansen]).

[0157] If y is a zero of f, then f(y)=0, and (21) is replaced by,

f(x)+J(i x, x^(I))(y−x)=0.  (22)

[0158] Define the solution set of (22) to be

s={y|ƒ _(i)(x)+[J(x, x′(i))(y−x)]_(i)=0,x′(i)εx ^(I)(i=1, . . . , n)}.

[0159] This set contains any point y ε x^(I) for which f(y)=0.

[0160] The smaller the box x^(I), the smaller the set s. The object ofan interval Newton method is to reduce x^(I) until s is as small asdesired so that a solution point y ε x^(I) is tightly bounded. Note thats is generally not a box.

[0161] Normally, the system of linear equations in (22) is solved usingany of a variety of interval methods. In the present situation if n<m,the linear system is overdetermined and therefore appears to beinconsistent. This is not necessarily the case. If the proceduredescribed above for interval linear equations is used to compute aninterval bound y^(I) on y, then y^(I) contains the set of consistentsolutions s.

[0162] For the solution of (22), the standard and distinctive notationN(x, x^(I)) is used in place of y^(I). This emphasizes the solution'sdependence on both x and x^(I).

[0163] From (22), define an iterative process of the form

f(x)+J(x, x ^(I))(N(x ^((k)) ,x ^(I(k)))−x)=0  (23a)

x ^(I(k+1)) =x ^(I(k)) ∩N(x ^((k)) , x ^(I(k)))  (23b)

[0164] for k=0,1,2, . . . where x^((k)) must be in x^(I(k)). A goodchoice for x^((k)) is the center m(x^(I(k))) of x^(I(k)). For details oncomputing N (x^((k)), x^(I(k))) when the system of interval linearequations appears to be overdetermined, see [Hansen].

[0165] Summary of Parameter Estimation in Nonlinear Models

[0166]FIG. 8 presents a flow chart summarizing the process of computinginterval parameter bounds from fallible measurements in accordance withan embodiment of the present invention. During operation, the systemreceives a set of measurements z₁, . . . , z_(n) (step 802), as well asvalues for measurement conditions c₁, . . . , c_(n) under which thecorresponding observations z_(i) were made (step 804). Next, the systemforms a system of interval nonlinear equations z_(i)−h(x|c_(i))+ε^(I)(x,c_(i))=0 (i=1, . . . , n) based on a nonlinear model h and an errormodel ε (step 806).

[0167] The system then uses standard interval Newtown techniquesdescribed above to solve the system of nonlinear equations to determineinterval parameter bounds for x. More specifically, the systemlinearizes the system of nonlinear equations (step 808), and then solvesthe system of linear equations using the technique described above withreference to FIG. 6 (step 810). The system then intersects the solutionwith the given box (step 812). Next, the system determines if thesolution has converged to be within specified tolerances (step 814). Ifso, the system stops. Otherwise, the system applies the interval Newtonprocedure for splitting if needed (step 816) and returns to step 808 tolinearize the system of equations again.

[0168] Conclusion for Parameter Estimation in Nonlinear Models

[0169] Computing bounds on nonlinear parameters from fallibleobservations is a pervasive problem. In the presence of uncertainobservations, attempting to capture uncertainty with Gaussian errordistributions is problematic when nonlinear functions of observationsare computed.

[0170] The procedure described in this specification uses the intervalsolution of a system of nonlinear equations to compute bounds onnonlinear parameters from fallible data. Among the many advantages ofthis approach is the ability to aggregate data from independentexperiments, thereby continuously narrowing interval bounds. Wheneverdifferent interval results are inconsistent, or if the set of intervalbounds from a given data set is empty, this proves an assumption isviolated or the model for the observations is measurably wrong.

[0171] Narrower parameter bounds can be computed from a calibratedsystem. It is interesting to note that the same procedure as describedabove can be used to solve the calibration problem. All that must bedone is to modify (8) in the following way:

[0172] replace selected unknown parameter values with their now measuredbounds x^(I) _(i); and

[0173] solve for narrower bounds on any parameters in the model forε^(I)(x, c_(i)).

[0174] The foregoing descriptions of embodiments of the presentinvention have been presented for purposes of illustration anddescription only. They are not intended to be exhaustive or to limit thepresent invention to the forms disclosed. Accordingly, manymodifications and variations will be apparent to practitioners skilledin the art. Additionally, the above disclosure is not intended to limitthe present invention. The scope of the present invention is defined bythe appended claims.

What is claimed is:
 1. A method for computing interval parameter boundsfrom fallible measurements, comprising: receiving a set of measurementsz₁, . . . , z_(n), wherein an observation model describes each z_(i) asa function of a p-element vector parameter x=(x₁, . . . , x_(p));storing the set of measurements z₁, . . . , z_(n) in a memory in acomputer system; forming a system of nonlinear equations z_(i)−h(x)=0(i=1, . . . , n) based on the observation model; and solving the systemof nonlinear equations to determine interval parameter bounds on x. 2.The method of claim 1, wherein the system of nonlinear equations is an“overdetermined system” in which there are more equations than unknowns.3. The method of claim 1, wherein each measurement z_(i) is actually aq-element vector of measurements z_(i)=(z_(il), . . . , z_(iq))^(T), andh is actually a q-element vector of functions h=(h₁, . . . , h_(q))^(T).4. The method of claim 1, wherein receiving the set of measurementsinvolves receiving values for a set of conditions c₁, . . . , c_(n)under which the corresponding observations z_(i) were made; and whereinequations in the system of nonlinear equations account for theconditions c_(i) and are of the form z_(i)−h(x|c_(i))=0 (i=1, . . . ,n).
 5. The method of claim 4, wherein each condition c_(i) is actuallyan r-element vector of conditions c_(i)=(c_(il), . . . , c_(ir))^(T). 6.The method of claim 4, wherein each condition c_(i) is not knownprecisely but is contained within an interval c^(I) _(i).
 7. The methodof claim 4, wherein equations in the system of nonlinear equations areof the form z_(i)−h(x|c_(i))+ε^(I)(x, c_(i))=0 (i=1, . . . , n), whichincludes an error model ε^(I)(x, c_(i)) that provides interval bounds onmeasurement errors for z_(i).
 8. The method of claim 7, wherein if z_(i)is actually a q-element vector of measurements z_(i)=(z_(il), . . . ,z_(iq))^(T), then ε^(I) is actually a q-element vector ε^(I)=(ε₁, . . ., ε_(q))^(T).
 9. The method of claim 7, wherein if there exists nosolution to the system of nonlinear equations, the method furthercomprises determining that at least one of the following is true: atleast one of the set of measurements z_(i), . . . , zhd n l is faulty;the observation model h(x|c_(i)) is false; the error model ε^(I)(x,c_(i)) is false; and the computational system used to compute intervalbounds on elements of x is flawed.
 10. The method of claim 1, whereinsolving the system of nonlinear equations involves: linearizing thesystem of nonlinear equations to form a corresponding system of linearequations; and solving the system of linear equations.
 11. The method ofclaim 10, wherein solving the system of nonlinear equations involvesusing Gaussian Elimination.
 12. A computer-readable storage mediumstoring instructions that when executed by a computer cause the computerto perform a method for computing interval parameter bounds fromfallible measurements, the method comprising: receiving a set ofmeasurements z₁, . . . , z_(n), wherein an observation model describeseach z_(i) as a function of a p-element vector parameter x=(x₁, . . . ,x_(p)); storing the set of measurements z₁, . . . , z_(n) in a memory ina computer system; forming a system of nonlinear equations z_(i)−h(x)=0(i=1, . . . , n) based on the observation model; and solving the systemof nonlinear equations to determine interval parameter bounds on x. 13.The computer-readable storage medium of claim 12, wherein the system ofnonlinear equations is an “overdetermined system” in which there aremore equations than unknowns.
 14. The computer-readable storage mediumof claim 12, wherein each measurement z_(i) is actually a q-elementvector of measurements z_(i)=(z_(il), . . . , z_(iq))^(T), and h isactually a q-element vector of functions h=(h₁, . . . , h_(q))^(T). 15.The computer-readable storage medium of claim 12, wherein receiving theset of measurements involves receiving values for a set of conditionsc₁, . . . , c_(n) under which the corresponding observations z_(i) weremade; and wherein equations in the system of nonlinear equations accountfor the conditions c_(i) and are of the form z_(i)−h(x|c_(i))=0 (i=1, .. . , n).
 16. The computer-readable storage medium of claim 15, whereineach condition c_(i) is actually an r-element vector of conditionsc_(i)=(c_(il), . . . , c_(ir))^(T).
 17. The computer-readable storagemedium of claim 15, wherein each condition c_(i) is not known preciselybut is contained within an interval c^(I) _(i).
 18. Thecomputer-readable storage medium of claim 15, wherein equations in thesystem of nonlinear equations are of the form, z_(i)−h(x|c_(i))+ε^(I)(x,c_(i))=0 (i=1, . . . , n), which includes an error model ε^(I)(x, c_(i))that provides interval bounds on measurement errors for z_(i).
 19. Thecomputer-readable storage medium of claim 18, wherein if z_(i) isactually a q-element vector of measurements z_(i)=(z_(il), . . . ,z_(iq))^(T), then ε^(I) is actually a q-element vector ε^(I)=(ε_(l), . .. , ε_(q))^(T).
 20. The computer-readable storage medium of claim 18,wherein if there exists no solution to the system of nonlinearequations, the method further comprises determining that at least one ofthe following is true: at least one of the set of measurements z_(i), .. . , z_(n) is faulty; the observation model h(x|c_(i)) is false; theerror model ε^(I)(x, c_(i)) is false; and the computational system usedto compute interval bounds on elements of x is flawed.
 21. Thecomputer-readable storage medium of claim 12, wherein solving the systemof nonlinear equations involves: linearizing the system of nonlinearequations to form a corresponding system of linear equations; andsolving the system of linear equations.
 22. The computer-readablestorage medium of claim 21, wherein solving the system of nonlinearequations involves using Gaussian Elimination.
 23. An apparatus thatcomputes interval parameter bounds from fallible measurements,comprising: a receiving mechanism configured to receive a set ofmeasurements z₁, . . . , z_(n), wherein an observation model describeseach z_(i) as a function of a p-element vector parameter x=(x₁, . . . ,x_(p)); a memory in a computer system for storing the set ofmeasurements z₁, . . . , z_(n); an equation forming mechanism configuredto form a system of nonlinear equations z_(i)−h(x)=0 (i=1, . . . , n)based on the observation model; and a solver configured to solve thesystem of nonlinear equations to determine interval parameter bounds onx.
 24. The apparatus of claim 23, wherein the system of nonlinearequations is an “overdetermined system” in which there are moreequations than unknowns.
 25. The apparatus of claim 23, wherein eachmeasurement z_(i) is actually a q-element vector of measurementsz_(i)=(z_(il), . . . , z_(iq))^(T), and h is actually a q-element vectorof functions h=(h₁, . . . , h_(q))^(T).
 26. The apparatus of claim 23,wherein the receiving mechanism is additionally configured to receivevalues for a set of conditions c₁, . . . , c_(n) under which thecorresponding observations z_(i) were made; and wherein equations in thesystem of nonlinear equations account for the conditions c_(i) and areof the form z_(i)−h(x|c_(i))=0 (i=1, . . . , n).
 27. The apparatus ofclaim 26, wherein each condition c_(i) is actually an r-element vectorof conditions c_(i)=(c_(il), . . . , c_(ir))^(T).
 28. The apparatus ofclaim 26, wherein each condition c_(i) is not known precisely but iscontained within an interval c^(I) _(i).
 29. The apparatus of claim 26,wherein equations in the system of nonlinear equations are of the formz_(i)−h(x|c_(i))+ε^(I)(x, c_(i))=0 (i=1, . . . , n), which includes anerror model ε^(I)(x, c_(i)) that provides interval bounds on measurementerrors for z_(i).
 30. The apparatus of claim 29, wherein if z_(i) isactually a q-element vector of measurements z_(i)=(z_(il), . . . ,z_(iq))^(T), then ε^(I) is actually a q-element vector ε^(I)=(ε₁, . . ., ε_(q))^(T).
 31. The apparatus of claim 29, wherein if there exists nosolution to the system of nonlinear equations, the solver is configuredto determine that at least one of the following is true: at least one ofthe set of measurements z_(i), . . . , z_(n) is faulty; the observationmodel h(x|c_(i)) is false; the error model ε^(I)(x, c_(i)) is false; andthe computational system used to compute interval bounds on elements ofx is flawed.
 32. The apparatus of claim 23, wherein the solver isconfigured to: linearize the system of nonlinear equations to form acorresponding system of linear equations; and to solve the system oflinear equations.
 33. The apparatus of claim 32, wherein the solver isconfigured to solve the system of nonlinear equations using GaussianElimination.